A. Ratio B. Simplifying ratio Before simplifying a ratio any units should be made the same a. 15 : 20 3 : 4 A ratio is used to compare quantities 2 : 1 means the quantity on the left is double the quantity on the right 1 : 3 means the quantity on the right is treble the quantity on the left 4 : 3 for every 4 on left there is 3 on the right A ratio has no units A simplified ratio only uses whole numbers 5 b. 50p : 3 50 : 300 1 : 6 c. 2kg : 400g 2000 : 400 20 : 4 5 : 1 50 100 4 1kg = 1000g d. 8 : 3.2 80 : 32 10 : 4 5 : 2 x10 8 2 Firstly multiply to make the decimal a whole number e. 5 6 : 2 3 = 5 6 : 4 6 = 5 : 4 Firstly make equivalent fractions with the same denominator C. Ratio and fractions a. James and Elaine share a box of sweets in the ratio 2 : 3. What fraction of the sweets do they each receive? 2 : 3 means there is 5 parts so James gets 2 5 and Helen gets 3 5 of the sweets b. An orange drink is made by mixing 2 parts juice with 5 parts water. This means 5 7 of the drink is water and 2 7 is juice D. The ratio 1 : n or n : 1 Just divide to make one side = 1 This can make it easier to compare quantities a. 4 : 12 1 : 3 4 We can now see the right is 3 times the size of the left b. 4 : 80p 400 : 80 40 : 8 5 : 1 10 8 The left is 5 times the size of the right c. 9 : 2 4.5 : 1 e. 500g : 2kg 500 : 2000 1 : 4 d. 5 : 9 2 5 The left is 4.5 times the size of the right The right is 4 times the size of the left or 1 : 9 5 1 : 1.8 500g : 2kg 500 : 2000 500 2000 : 1 = 1 4 : 1 The right is 1.8 times the size of the left The left is one quarter the size of the right www.teachitmaths.co.uk 2017 28629 Page 1 of 8
E. Dividing into a ratio Read the questions carefully as not all questions need the same method a. 60 is to be share between Gary and Susan in the ratio 2 : 3. How much does Susan receive? 60 2 : 3 needs 5 parts (2 + 3 = 5) = 12 in each part 5 Susan receives 3 x 12 = 36 b. Split 80 into two groups where one group is 3 times as big as the other 80 = 20 in each part 3 x 20 = 60 4 3 : 1 or 1 : 3 Needs 4 parts 60 : 20 or 20 : 60 c. Carol and Tim received some money in the ratio 5 : 4. Tim received 60, how much was Carol's share? C : T This time we know or Tim got 60 = 4 parts 5 : 4 how much one x15 1 part = 60 4 = 15 person receives? : 60 Carol got 5 x 15 = 75 5 x 15 = 75 The multiplier is 60 4 = 15 d. To make concrete Alan mixes sand, cement and gravel in the ratio 3 : 1 : 4. He has 10kg of gravel. How much sand and cement should he use? C : S : G 3 : 1 : 4 x2.5 The multiplier is 10 4 = 2.5? :? : 10 Cement = 3 x 2.5 = 7.5kg Sand = 1 x 2.5 = 2.5kg e. Helen and Ann shared a box of sweets in the ratio 4 : 7. Ann got 12 more sweets than Helen. How many sweets did Ann have? This time we know the gap H : A The gap = 7 4 = 3 4 : 7 3 parts = 12 sweets 1 part = 12 3 = 4 sweets Ann has 7 x 4 = 28 sweets We could use a build up method The left increases in steps of 4 The right increases in steps of 7 H : A 4 : 7 (gap = 3) 8 : 14 (gap = 6) 12 : 21 (gap = 9) 16 : 28 (gap = 12) Ann has 28 sweets Keep going until the gap is 12 www.teachitmaths.co.uk 2017 28629 Page 2 of 8
F. Direct proportion When quantities are in direct proportion their ratio stays the same as they increase or decrease. e.g. If one doubles so does the other a) 4 books cost 15 Find the cost of 8 books 15 x 2 = 30 Find the cost of 2 books 15 2 = 7.50 Find the cost of 20 books 5 x 15 = 75 Find the cost of 1 book 15 4 = 3.75 Now we know the cost of 1 book we can calculate the cost of any number of books Find the cost of 7 books 7 x 3.75 = 26.25 b) 6 boxes weigh 18kg, how much do 9 weigh? 18 x 1.5 = 27kg or 18 + 9 = 27kg Or find the weight of 1 box first Half as much again 1 box weighs 18 6 = 3kg 9 boxes weigh 9 x 3kg = 27kg c) 5 tickets cost 45, how much do 8 tickets cost? 1 ticket costs 45 5 = 9 8 tickets cost 8 x 9 = 72 It s not easy to turn 5 into 8. So find the cost of 1 ticket first d) Brian is going to the USA and sees the exchange rate 1 = $1.37 (i) How much will he get changing 250 into US dollars 250 x 1.37 = $342.50 Every 1 gives $1.37 (ii) He returns home with $42. How much will he get in pounds when he converts this? $42 1.37 = 30.65693431 = 30.66 Every $1.37 gives 1 Remember to use correct money notation e) Matthew is on holiday in Spain and sees a new mobile phone on sale for 165. He knows he can buy it at home for 140. Will the phone be cheaper in Spain or at home? The exchange rate is 1 = 1.24 140 x 1.24 = 173.60 so it is cheaper in Spain or 165 1.24 = 133.0645161 = 133.06 so it is cheaper in Spain Convert the Euro price to or the price to Euro www.teachitmaths.co.uk 2017 28629 Page 3 of 8
G. Inverse proportion When quantities are in invese proportion as one increases the other decreases at the same rate. a. It takes 1 man 4 days to decorate a room. How long will it take 2 men? Double the men so half the time 4 2 = 2 days b. It takes 50 men 1 hour 40 minutes to erect a stage. How long will it take 40 men? 1hr 40 = 100 mins Be careful if using a calculator: 1hr 40m is not 1.4 hrs (Using minutes will be easier) 50 men = 100 mins 10 men = 5 x 100 = 500 mins (fewer men will take longer) 40 men = 500 4 = 125 mins = 2hr 5 mins (more men will take less time) c. It takes 4 men 3 hours to lay a 600m 2 lawn in a garden. How long will it take 5 men to lay a 1000m 2 lawn? Change one variable at a time. When you change one variable only alter one of the other variables Men Time Lawn 4 men 3 hours 600m 2 STARTING information 1 man 12 hours 600m 2 (fewer men will take more time) 1 man 2 hours 100m 2 (less lawn will take less time) 5 men 2 hours 500m 2 (more lawn will take more men) 5 men 4 hours 1000m 2 (more lawn will take more time) It will take them 4 hours An alternative route remember changing one variable will affect only one other Men Time Lawn 4 men 3 hours 600m 2 STARTING information 4 men 0.5 hours 100m 2 (less lawn will take less time) 1 man 2 hours 100m 2 (fewer men will take more time) 1 man 20 hours 1000m 2 (more lawn will take more time) 5 men 4 hours 1000m 2 (more men will take less time) It will take them 4 hours www.teachitmaths.co.uk 2017 28629 Page 4 of 8
H. Value for money To find the best buy or the best value for money we need to compare quantities of the same size a. Bubble bath is available in two sizes. Which size offers best value? 100ml 1.65 250ml 4.15 Method 1 (Make the small size up to the larger size) 250 = 2.5 (The multiplier is 2.5) 100 1.65 x 2.5 = 4.125 = 4.13 Smaller bottle is better value Method 2 (Make a new quantity for both bottles) 5 small bottles cost 5 x 1.65 = 8.25 2 large bottles cost 2 x 4.15 = 8.30 Smaller bottle is better value 100ml and 250ml Can both be made into 500ml Method 3 (Find how much you can buy for 1p) 100ml = 165p 250ml = 415p 100 165 ml = 1p 250 ml = 1p 415 0.606 ml = 1p 0.602ml = 1p Smaller bottle is better value You get more for 1p in the smaller bottle 1 ml is cheaper in the smaller bottle Method 4 (Find the price of 1ml for both bottles) 100ml = 165p 1 ml = 165 100 250ml = 415p 1 ml = 415 250 1 ml = 1.65p 1 ml = 1.65p Smaller bottle is better value Put units on your working out. It reminds you what you have calculated so you can make the correct decision www.teachitmaths.co.uk 2017 28629 Page 5 of 8
I. Direct proportion using algebra When two quantities are in direct proportion one is a multiple of the other Eg y = 3x p = 5q f = t 7 a = 1 3 c Learn the method to solve a direct proportion problem 1. Write a starting statement eg y α x or y α x 2 etc 2. Change this to an equation eg y = kx or y = kx 2 (We know one is a multiple of the other, k is the value of the multiplier) 3. Use the known facts to calculate k 4. Rewrite the correct using equation using your value of k 1. y is directly proportional to the square of x, and when x = 4, y = 48 (i) Write an equation expressing y in terms of x (ii) Find the value of y when x = 6 (iii) Find the value of x when y = 18.75 (i) y α x 2 (the starting statement) y = kx 2 (the equation) 48 = k x 4 2 48 = k x 16 48 16 = k k = 3 y = 3x2 We know it is direct proportion and we know it is the square of x Now we know the equation we can use it to answer the other questions (ii) y = 3x 2 y = 3 x 6 2 y = 3 x 36 y = 108 (iii) y = 3x 2 18.75 = 3x 2 18.75 = x 2 3 x 2 = 6.25 x = 6.25 x = 2.5 www.teachitmaths.co.uk 2017 28629 Page 6 of 8
J. Inverse proportion using algebra When two quantities are in inverse proportion if as one increases the other decreases Eg y = 1 x 24 = pq b = 5 c Learn the method to solve an inverse proportion problem 1. Write a starting statement eg y α 1 x y α 1 x 2 etc 2. Change this to an equation eg y = k x or yx = k 3. Use the known facts to calculate k y = k x 2 or yx2 = k 4. Rewrite the correct using equation using your value of k 1. T is inversely proportional to the cube of p, and when p = 2, T = 5 (i) Write an equation expressing T in terms of p (ii) Find the value of T when p = 5 (iii) Find the value of p when T = 0.625 We know it is inverse proportion so it is 1 over and we know it is the cube of p (i) T α 1 (the starting statement) p3 T = k (the equation) p3 5 = k 2 3 5 = k 8 5 x 8 = k k = 40 T = 40 3 or Tp 3 = 40 (i) T α 1 p 3 Tp 3 = k 5 x 2 3 = k 5 x 8 = k k = 40 Tp 3 = 40 or T = 40 p 3 An alternative method (ii) T = 40 p 3 T = 40 5 3 T = 40 125 T = 0.32 (iii) T = 40 p 3 0.625 = 40 p 3 p 3 40 = 0.625 p 3 = 64 p = 3 64 p = 4 www.teachitmaths.co.uk 2017 28629 Page 7 of 8
K. Map scales Do you understand 1 : 100 000 or 1 : 50 000? 1 : 100 000 means for every 1 on the map there is 100 000 in real life. As ratio does not use units, it can be 1 cm = 100 000 cm or 1 mm = 100 000 mm etc 1. A map uses a scale of 1 : 100 000. How long is the road that is 4.3cm on the map? 4.3cm x 100 000 = 430 000cm = 4300m = 4.3km 100cm = 1m 1000m = 1km 2. A road is 6.4km long. How long will it be on a map that uses a scale of 1 : 50000? 6.4km = 6400m = 640000cm 640000 50000 = 12.8cm = 4.3km Convert 6.4km to cm 3. A lake appears as 3.5cm 2 on a map with a scale of 1 : 50 000. What is the area of the real lake? You need to remember: Large length = Small length x SF Large area = Small area x SF 2 Large volume = Small volume x SF 3 1m 1m 1m 2 100cm Large area = Small area x SF 2 Large area = 3.5 x 50 000 2 = 3.5 x 50 000 x 50 000 = 8750000000 cm 2 8750000000 = 875000 m 2 10000 = 0.875 km 2 100cm 1m 2 = 100 x 100 = 10000cm 2 Can you expain why it is 0.875km 2 www.teachitmaths.co.uk 2017 28629 Page 8 of 8